Heat a large well-seasoned cast iron skillet over medium-low heat.
Lay the bread slices on a work surface and spread about a teaspoon of mustard on each slice. Divide the sliced cheese among all 8 slices. Evenly spread the meat over 4 cheese-topped bread slices, followed by 1/4 of the arugula. Season with salt and pepper and cover with the remaining 4 slices of bread.
Working in 2 batches, if necessary, put 2 of the sandwiches into the skillet. Weight the sandwiches down with a bacon press or cover them with a sheet of parchment and set a slightly smaller skillet directly on top. Put a can of tomatoes or a brick in the top skillet to weigh the pan down. Cook until the sandwich bottoms are golden and cheese is beginning to melt, 5 to 6 minutes. Remove the bacon press or top skillet, flip the sandwiches, return the press or parchment and top skillet to the pan and cook until golden brown, an additional 3 to 4 minutes. Repeat with the remaining 2 sandwiches.
Slice the panini on a diagonal and arrange on a serving platter. Serve immediately.